On the stability of φ-uniform domains [On the stability of phi-uniform domains]

Abstract

<p> We study two metrics, the quasihyperbolic metric and the distance ratio metric of a subdomain G &sub; ℝ. In the sequel, we investigate a class of domains, so called &phi;-uniform domains, defined by the property that these two metrics are comparable with respect to a homeomorphism &phi; from [0,&infin;) to itself. Finally, we discuss a number of stability properties of &phi;-uniform domains. In particular, we show that the class of &phi;-uniform domains is stable in the sense that removal of a geometric sequence of points from a &phi;-uniform domain yields a &phi;-uniform domain. &copy; 2013 Springer-Verlag Wien.</p>